Mechanical timers came into use for chess tournaments in the late 1800's, replacing sandglasses. A chess clock, as it is now called, actually comprises a pair of clocks running alternately. Each clock is controlled by a switch, usually in the form of a pushbutton, which stops the clock on the side of the player who is completing a move and simultaneously starts the clock of the player whose turn it is to move. To start a game the player with the black pieces (Black) pushes the move button on his/her side, thereby starting the clock of the player with the white pieces (White), who makes the first move of the game. Each clock by this means records the cumulative time taken by each player in the sequence of his/her moves. The players are required to complete a specified number of moves within an initially allotted time, known as a time control, on pain of forfeit. There may be a plurality of time controls, basically subsequences of the sequence of moves by each player, with a varying number of required moves and time periods. The overall timing scheme is herein called a time limit, although usage varies.
The traditional time limit known as “forty in two” will serve as an example. It includes a primary time control of 40 moves in two hours, a secondary time control of 20 moves in one hour, and sometimes a tertiary time control of 10 moves in 30 minutes. The final time control in the series is repeated to the end of the game. Time left over on a player's clock at the end of a time control is added to his/her next time control. Players are usually responsible for keeping track of the number of moves completed by recording moves in chess notation.
The speed at which moves are made may vary considerably from long, drawn out moves in a complex position to split-second moves in time trouble. The playing speed enforced by a time control is called an average speed in mathematics, which may be writtenS=M/T,  (1)where M is the required number of moves and T is the allotted time. S is in terms of moves per hour or, for speed chess, moves per minute. It is clear from this formula that if a player fails to complete the required number of moves in the allotted time, his/her average playing speed when the forfeit occurs will be less than the minimum imposed by the time control. Hereafter in this application for the sake of brevity playing speed may be taken to mean minimum average playing speed. In the time limit of the example above, the designation “forty in two” adequately describes the requirements of the first time control. It is easily established that the playing speed is 20 moves per hour and, furthermore, that the playing speed of the second and third time controls is also 20 moves per hour.
The above mathematical description of time controls is not current in chess circles, where such controls are regarded primarily as a means of expediting play. “Forty in two,” for example, is rarely described as 20 moves per hour. Playing speed in general has come to be a neglected concept with the advent of digital time limits, especially because it is now more difficult to calculate it with precision.
Advances in Timing Technology
In the 1970's electronic digital chess clocks (usually powered by batteries) emerged. The first commercially available digital chess clock was patented by Joseph Meshi in 1975 (U.S. Pat. No. 4,062,180). Digital chess clocks make possible variations in timing, in particular, delays or increments, intended to make enforcement of time controls more flexible. Time controls continued to be described in terms of a required number of moves over a period of time. Delays and increments were considered negligible in time controls, and it was not generally recognized that they affect playing speed. Before describing these effects, it will be useful to review the three major delay schemes that are enabled by digital clocks.
The Fischer Clock
The problem of time scrambles (i.e., frantic attempts to avoid a forfeit on time) was addressed by a 1988 invention of Robert J. Fischer (U.S. Pat. No. 4,884,255), known as the Fischer Clock. The mechanism of the Fischer Clock adds a predetermined amount of time, typically one or two minutes, to a player's remaining time for every move that he/she completes. The additional time, called a bonus or increment, was intended as a reward for completing moves with dispatch. As Fischer pointed out in his patent application:                If a 2-minute interval is selected, a player will always have [the original time] remaining on his clock if that player spends exactly 2 minutes on every single move during the game. For each move that a player spends less than 2 minutes, he will accumulate additional time on his clock for use on future moves. For each move that a player spends more than 2 minutes, a player will use up some of his stored up time from either previous moves or from his initial time period.This scheme usually has the intended effect of avoiding time scrambles, but the effect on playing speed was never made clear.Time Delays        
Another method for avoiding time scrambles is embodied in digital game timers with a move delay (U.S. Pat. No. 5,420,850 to Cameratta et al., 1995). When a player completes his/her move on such a timer, the opponent's clock does not start immediately. There is instead a slight delay, typically five seconds, considered free thinking time for the player on the move. The idea is that a player will always have, at a minimum, the period of the delay to complete his/her move. This delay, like the Fischer increment, allows extra time for deliberation. However, the Fischer increment is added after a move is completed while the simple delay occurs at the start of a move. A simple delay never causes time to be added to a clock display. Any unused portion of the delay when a player moves quickly is simply discarded.
Time delays over the course of a game distort the intended playing speed. Official rules of the United States Chess Federation (USCF) once provided that “the tournament director has the right to shorten the basic time control” in compensation for move delays (Fifth Edition, Rule 5Fa). However, this rule has recently been deleted: “All clocks, including digital, delay, analog, are to be set for the same base (starting) time” (www.uschess.org, Rulebook Changes, 2003-2014). The effect of move delays on playing speed, in short, is to be ignored.
The Bronstein Clock
The oldest of the timing schemes was introduced by Soviet grandmaster David Bronstein (1924-2006). It is like the Fischer Clock in that time is added to a player's clock at the completion of each move, but with the Bronstein Clock the time added is never greater than the time spent on a move. Consequently, there may be no net change on the clock at the completion of a move, and one does not see the expanding time control that often occurs in the Fischer Clock. The Bronstein delay (or increment) is similar in effect to a simple delay, and USCF rules consider the two equivalent (Rule 5F: Tip). However, the fact that the Bronstein delay comes after the move, while the simple delay occurs at its beginning, requires different handling at the beginning and end of a time control. Once again the primary concern is to avoid time scrambles. There has been no analysis, to my knowledge, of the precise effect on playing speed.
Displaying Playing Speed
The delays and increments which modify the operation of a game timer were primarily intended as means of dealing with time scrambles. It was never considered necessary to calculate their timing effects precisely, and the fact that they do not alter initially allotted time reinforced this impression. Modifications of timing nevertheless have an effect on the total elapsed time in a time control, denoted in Formula 1 as T, and consequently on playing speed, S. (Such modifications will hereafter simply be termed delays since both delays and increments have the effect of augmenting total time.) Chess clocks display the time remaining on each side and sometimes (in digital versions) the number of moves remaining in a time control. But playing speed is an essential aspect of time limits, and its display would be a valuable feature in modern timers.
Sudden Death
The tendency to neglect playing speed in chess clocks was no doubt exacerbated by the introduction in recent years of a time control known as sudden death. A sudden-death time control specifies a period of time in which a game is to be completed, that is, either an entire game (Game/SD) or the portion remaining after one or more normal time controls. The number of moves required is indeterminate. Since a sudden-death control, if one is included in a time limit, must be the last in a series of time controls, it may be said to govern a terminal subsequence of moves. It thus facilitates the scheduling of games in large amateur tournaments, but it makes the precise calculation of playing speed difficult. There is, however, a way to estimate playing speed in sudden death, at least for Game/SD, as will be shown.
When there is a definite number of moves required in a time control, it governs a nonterminal subsequence of moves. In the absence of a sudden-death control, the last time control in a series of nonterminal controls repeats itself indefinitely with respect to the period of time and the number of moves in the subsequence.
Playing Speed as Input
In a previous invention by this applicant (U.S. Pat. No. 7,887,232, issued 2011) minimum average playing speed was presented as an input value. Playing speed was entered by the user into the timer, along with other values, to obtain a starting time. A problem with this approach is that playing speed is not ordinarily dealt with in chess practice, and it is a value that in many cases would not be easy for the average chess player to calculate.